Daniel's triple challenge

Algebra Level 5

How many ordered triples of complex numbers (a,b,c)(a, b, c) are there such that a3ba^3-b, b3cb^3-c, and c3ac^3-a are rational numbers, and

a2(a4+1)+b2(b4+1)+c2(c4+1)=2(a3b+b3c+c3a)?a^2(a^4+1)+b^2(b^4+1)+c^2(c^4+1)=2(a^3b+b^3c+c^3a)?

This problem is posed by Daniel C.

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